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THE AUSTRALIAN NATIONAL UNIVERSITY
WORKING PAPERS IN ECONOMICS AND ECONOMETRICS
PUBLIC CAPITAL SPILLOVERS AND GROWTH: A FORAY DOWNUNDER
Timothy Kam and Yi-Chia Wang *
School of Economics
College of Business and Economics The Australian National University
Working Paper No. 474
ISBN 086831 474 9
September 2006
ABSTRACT. We extend the deterministic growth model of Glomm and Ravikumar (1994) to a stochastic endogenous growth model which nests both exogenous and endogenous growth factors. By introducing simple shocks to production technology, private capital and public capital investment, we can derive testable time series properties of the analytical model. The hypothesis of strict endogenous growth due to public capital spillovers cannot be statistically rejected for our Australian data set. We find further short-run evidence of public capital contributing to permanent increases in the levels of per capita income and private capital. * Corresponding author: TIMOTHY KAM, Email: [email protected] KEYWORDS: Public capital spillover; stochastic growth; time series J.E.L. CODE: O41; C32
ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474 1
PUBLIC CAPITAL SPILLOVERS AND GROWTH: A FORAY DOWNUNDER
BY TIMOTHY KAM AND YI-CHIA WANG
ABSTRACT. We extend the deterministic growth model of Glomm and Ravikumar (1994) to astochastic endogenous growth model which nests both exogenous and endogenous growth factors.By introducing simple shocks to production technology, private capital and public capital invest-ment, we can derive testable time series properties of the analytical model. The hypothesis ofstrict endogenous growth due to public capital spillovers cannot be statistically rejected for ourAustralian data set. We find further short-run evidence of public capital contributing to permanentincreases in the levels of per capita income and private capital.
KEYWORDS: Public capital spillover; stochastic growth; time series
J.E.L. CODE: O41; C32
1. INTRODUCTION
IN THIS PAPER, we extend the deterministic growth model of Glomm and Ravikumar
(1994) to a stochastic growth version with endogenous public capital spillovers. By
introducing simple shocks to production technology, private capital and public capital in-
vestment, we can derive testable time series properties of the analytical model along the lines
of Lau and Sin (1997) who first investigated a similar question for the US. We allow growth
of per capita income to be generated exogenously via Harrod-neutral technical progress
and/or endogenously by aggregate public infrastructure spillovers. The postulation of strict
endogenous growth is tested empirically for Australia using a constructed annual data set
for the period 1959/60–2003/04.To the best of our knowledge, there has been no work that takes the approach in Lau and
Sin (1997), of using a theory-consistent approach to test for endogenous growth effects, with
respect to Australian data. We show that the hypothesis of strict endogenous growth due to
public capital spillovers cannot be statistically rejected. Unlike Lau and Sin (1997) we also
explore the short-run causal links between public capital accumulation and growth. Since
the time-series causal chain running from public capital accumulation to private outcomes
is obvious in the theoretical model, we investigate whether this short-run effect is plausible
using a vector error correction model. The idea here is to let the data speak as much as
possible with respect to the short-run dynamics while imposing the estimated long-run
∗Corresponding author: TIMOTHY KAM, Email: [email protected]
ISBN 086831 474 9 c©-left 2006 Kam and Wang
2 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
relationship implied by the theory. We find further short-run evidence of public capital
contributing to permanent increases in the levels of per capita income and private capital.
The role of public capital as distinct from private capital in fostering growth has received
attention from the economics profession as early as Arrow and Kurz (1970). This hypothesis
became known in the empirical literature as the public capital debate, which began with
the seminal work of Aschauer (1989a). Aschauer’s method of estimating a single aggregate
production function (which incorporates public capital stock) was first adapted for Australian
studies by Otto and Voss (1994). Both papers found that there was a significantly large
elasticity of output (in the order of 0.40) with respect to public capital. Their methodology
was not without criticism. The critiques range from claims of possible endogeneity of the
public capital variable to the ad hoc nature of imposing a production function.1
Nevertheless, the production function approach is still valid and not ad hoc, albeit subject
to a different interpretation. The production function can be interpreted as a long run
relationship between output, and the private and public inputs, as in Flores de Frutos, Gracia-
Dıez, and Perez-Amaral (1998). Furthermore, this time series property of the variables can
be derived from a stochastic growth framework with sound microfoundations.
The paper is thus arranged. A stochastic growth version of the Glomm and Ravikumar
model and the time series (cointegration) basis of the production function framework is
derived in Section 2. Section 3 contains the estimation and test of strict endogenous growth
within the cointegrating relationship. The short-run and impulse response analysis using the
VEC(2) structure is enumerated in Section 4. The paper concludes with Section 5.
2. THE THEORETICAL MODEL
A stochastic growth version of the Glomm and Ravikumar model is presented in this
Section. A representative household-worker chooses an optimal consumption or investment
path to maximise expected lifetime utility, given resource constraints and taking government
policy as given. The fiscal policy is assumed to be a Ramsey planning problem subject to
technological constraints and a periodic balanced budget a la Barro (1990).
2.1 Technology and household choice
Let Y be aggregate output, K be aggregate private capital stock, L be the total number
of workers or population and G be a measure of congestion-adjusted public capital stock
to be defined later. The Harrod-neutral rate of technological progress is denoted by x. We
would like public capital to enter aggregate production so that potentially there would be
1See Sturm (1998, pp. 57-65) for a survey. See e.g. Lynde and Richmond (1992) and Berndt and Hansson (1991).
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 3
a spillover effect.2 Assume that the aggregate production function takes the Cobb-Douglas
form
Yt = AKαt
[(1 + x)tLt
]1−αGθt εPt ; α, θ ∈ (0, 1) (1)
which yields the production function in per worker terms as
yt = A(1 + x)(1−α)tkαt Gθt εPt ; α, θ ∈ (0, 1) (1’)
where the lower case variables, y and k, denote per worker output and private capital
respectively. Thus, the model nests the possibilities of exogenous and/or endogenous growth.
The production technology is subject to random shocks, εPt , assumed to be multiplicative in
this model. Aggregate public capital, Gt, enters as an input into production (implying the
spillovers or externality effect) and it is taken by the representative agent as given. Further,
aggregate public capital is subject to congestion from its use by private production
Gt =Gt
Kφt [(1 + x)tLt]
1−φ ; φ ∈ (0, 1) (2)
where Gt is the aggregate stock of public infrastructure investment and φ and (1−φ) denote
the degree of congestion arising from private capital stock and labor force, respectively. This
is contrary the usual notion that public goods are non-exclusive and non-rival.
We can further detrend equation (1’). Let yt := yt/(1 + x)t, kt := kt/(1 + x)t, and
gt := Gt/Lt(1 + x)t. Thus equation (1’) can be written in per efficiency unit worker terms
as
yt = Akα−θφt gθt εPt ; α, θ, φ, (α− θφ) ∈ (0, 1) (3)
Assume that there is 100 per cent depreciation at the end of each period for private capital.
Then private per capita investment will give the following period’s capital stock per efficiency
unit worker
kt+1 = itεKt+1 (4)
where i is investment per efficiency unit worker and εKt+1 is a random shock and k0 is given.
Similarly, aggregate public infrastructure investment is assumed to depreciate fully at the
end of the period
Gt+1 = IGt εGt+1 (5)
2This effect would depend on the parameters and is the object of empirical testing later.
4 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
where IG is aggregate public expenditure on infrastructure and G0 is given.Let τ be the uniform income tax rate. The household solves
V (k0, εP0 ) = max
{ct,kt+1}∞t=0
E0
∞∑t=0
βt ln(ct); β ∈ (0, 1) (6)
subject to
a) kt+1 = (1− τt)Akα−θφt gθt ε
Pt − ct
b) k0, g0 given
c) ct, kt+1 ≥ 0
for all t ∈ N.It is shown in Appendix A, by restating the problem in equation (6) as a dynamic program,
that the solution to the household problem taking government policy {τt, gt}∞t=0 as given,
yields the optimal paths of consumption and private capital as
ct = (1− τt) [1− (α− θφ)β)]Akα−θφt gθt εPt (7)
kt+1 = (1− τt)(α− θφ)βAkα−θφt gθt εPt (8)
for all states and dates t ∈ N, given k0. The analytical solutions were obtainable by assuming
logarithmic utility, Cobb-Douglas technology, 100 per cent depreciation of private and public
capital, a uniform tax structure and a balanced budget. This also simplifies the cointegrating
properties of the variables.
2.2 Public sector
The government budget is such that public investment demand each period is exactly
financed by income tax revenue:
IGt = τtYt (9)
We assume a government policy to be one that implements a Ramsey optimal fiscal plan.
The government maximizes the same objective function as households but it also takes into
account the optimal behavior of private agents with respect to the policy plan in a competitive
equilibrium. The optimal policy is determined by solving
v(k0, g0, εP0 ) = max
{τt,kt+1,gt+1}∞t=0
E0
∞∑t=0
βt ln{
(1− τt) [1− (α− θφ)β)]Akα−θφt gθt εPt
}(10)
subject to
a) τt ∈ (0, 1)
b) gt+1 = τtAkα−θφt gθt ε
Pt
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 5
c) kt+1 = (1− τt)(α− θφ)βAkα−θφt gθt εPt
d) k0, g0 given
for all t ∈ N. Notice that we have replaced per period consumption in the objective with
the competitive behavior in (7) and also encoded households’ optimal capital investment
decision (8) into constraint c) above.
It is assumed that the benevolent government maximises household welfare when it max-
imises household consumption growth. A further assumption is that the sequence {Gt}∞t=0
is bounded above by {ηtGt}∞t=0 for some value η ≥ 1, to ensure that the infinite horizon
household objective is bounded above for all feasible consumption paths. In other words,
the optimal paths in equations (7) and (8) will be unique: Glomm and Ravikumar (1994).
2.3 Optimal public policy
Solving the government’s problem by dynamic programming (Appendix B), it is found
that the optimal tax rate is a function of constants.Specifically, the optimal tax rate is defined
by the function
τt = θβ (11)
for all states and dates t ∈ N. Thus, the optimal tax rate is equal to the one-period discounted
share of public capital in output, where the government faces the same subjective discount
rate, β, as the household.
Second, given the optimal choice of public policy, the evolution of private capital per
efficiency unit worker in equation (8) can be described by the first-order stochastic difference
equation
kt+1 = (1− θβ)(α− θφ)βAkα−θφt gθt εPt (12)
The evolution of public capital per efficiency unit worker is
gt+1 = θβAkα−θφt gθt εPt (13)
Consequently, the ratio of the optimal paths for private and public capital stays constant
over time. This can be observed by taking the ratio of equation (13) to equation (12), which
yields
gt+1
kt+1
=θ
(1− θβ)(α− θφ)(14)
for all states and dates t ∈ N.
6 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
2.4 Long-run growth
Substitution of equation (14) into (12) gives the essential difference equation for the
evolution of private capital
kt+1 = [(1− θβ)(α− θφ)]1−θ θθβAkα+(1−φ)θt εPt (15)
Under an assumption of constant returns to scale to reproducible factors, where α + (1 −φ)θ = 1, the steady-state (εPt = 1; ∀t) growth rate of private capital will be given by
[(1− θβ)(α− θφ)]1−θ θθβA, which is perpetual and non-explosive. Also, output and public
capital will grow at the same rate as private capital, with constant returns to scale Cobb-
Douglas technology.
2.5 Testable time series properties of the model
Equations (12) and (13) can be written in natural logarithm and substitution of these into
the stochastic investment equations in (4) and (5) yields
ln kt+1 = ln [(1− θβ)(α− θφ)βA] + (α− θφ) ln kt + θ ln gt + ln εPt + ln εKt+1 (16)
and
ln gt+1 = ln (θβA) + (α− θφ) ln kt + θ ln gt + ln εPt + ln εGt+1 (17)
Multiplying equation (16) by (1 − θL) on both side, where L is the lag operator, and
substituting for (1 − θL) ln gt from equation (17) yields an equilibrium dynamic equation
of the log of per capita private capital expressed in terms of its own lags and the external
shocks
{1− [α+ (1− φ)θ]L} (ln kt + xt)
= {(1− θ) ln [(1− θβ)(α− θφ)βA] + θ ln(θβA)}
+ θL ln εGt + L ln εPt + (1 − θL) ln εKt (18)
Multiplying equation (17) by [1− (α− θφ)L] and substituting for [1− (α− θφ)L] ln kt from
equation (16) yields the equilibrium path for aggregate public capital
{1− [α+ (1− φ)θ]L} (ln gt − xt)
= {[1− (α− θφ)] ln(θβA) + (α− θφ) ln [(1− θβ)(α− θφ)βA]}
+ [1− (α− θφ)L] ln εGt + L ln εPt + (α − θφ)L ln εKt (19)
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 7
Also, taking logs of the equation for the private production function in equation (3), multi-
plying this by {1− [α+ (1− φ)θ)]L} and expressing this in per worker terms, yields
{1− [α+ (1− φ)θ]L} (ln yt − xt)
= {1− [α+ (1− φ)θ]} lnA+ (α− θφ) ln [(1− θβ)(α− θφ)βA] + θ ln(θβA)
+ (α − θφ) ln εKt + θ ln εGt + {1 + [α+ (1− φ)θ]L} ln εPt (20)
This equation describes the equilibrium path of the log of output per worker, ln yt.
Perpetual and stable growth at steady state: In this growth model, growth in per capita
output or income depends on the coefficient of the lagged output variable, α + (1 − φ)θ.
This is also the sum of all the exponents (or what is loosely known as the factor shares
in neoclassical terms) of the private and public inputs into production. There will be no
perpetual growth in the per capita variables once the economy reaches the steady-state path,
if α + (1 − φ)θ < 1, since the effects of past disturbances decay successively in equation
(20). Conversely, the steady-state growth path will be explosive if α+ (1− φ)θ > 1. In this
case there is increasing returns to all inputs.
To obtain perpetual growth with stability in the model, it is a requirement that α + (1−φ)θ = 1 and x = 0. This is the strict endogenous growth case. Thus, even if private
production displays diminishing returns to private inputs, overall it experiences constant
returns to scale due to the spillover effect from public capital. Hence there are two empirical
properties to be expected of the variables in the endogenous growth case. First, the sequences
{kt}∞t=0, {gt}∞t=0 and {yt}∞t=0 will be exact unit root processes. Second, and consequently,
the first difference of the logs of the per capita variables will be white noise processes, if
the linear combinations of the shocks in (18) to (20) are stationary in levels.
Derivation of cointegrating relationships: If there are three I(1) variables in the
system, there can be a maximum of two linearly independent cointegrating vectors. For
non-explosive, perpetual endogenous growth, it was concluded that α+(1−φ)θ = 1. Using
this fact in equations (19) and (20), and then subtracting the former from the latter, and
performing the same again on equation (18) and (20) gives the cointegrating space as
ln yt − ln kt = (1− L)−1 ln εPt − θ ln εKt + θεGt (21)
ln yt − ln gt = (1− L)−1 ln εPt + (1− θ) ln εKt − (1− θ)εGt (22)
If the cointegrating space in equation (21) and (22) is rejected, then there may be at most one
cointegrating vector. This cointegrating equation is a linear combination of all the variables.
This be shown by multiplying equation (21) on both sides by (1− θ), and and equation (22)
8 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
on both sides by θ, and then summing the two equations, to obtain
ln yt − (1− θ) ln kt − θ ln gt = (1− L)−1 ln εPt (23)
The cointegrating equation in (23) also represents the production function at steady state
with non-explosive, perpetual growth. In general, without assuming α + (1 − φ)θ = 1, the
single unrestricted cointegrating equation can be derived from equation (3) yielding
ln yt − (α− θφ) ln kt − θ ln gt − {1− [α+ (1− φ)θ]x}t− lnA = (1− L)−1 ln εPt (24)
Note that (23) is a nested case of (24) where (23) was derived under the hypothesis of
α + (1− φ)θ = 1. These possible cointegrating relationships will be tested in Section 3 of
this paper.
3. EVIDENCE FOR AUSTRALIA
3.1 Data
The empirical analysis in this part involves annual time series from 1959/60 to 2003/04
for Australia. It is important, for the purposes of testing for cointegrating relationships,
to have a longer series as opposed to a more frequently sampled series.3 Gross domestic
product (million AU$) at 2002/03 constant prices is constructed, with seasonal adjustment,
from Australian Bureau of Statistics (ABS) National Accounts, Table 5204.02. From ABS
National Accounts, Table 5204.70, we extracted both the year-end net government (public)
capital stock (million AU$) and year-end net private capital stock (million AU$). Both
types of net capital stock are generated by institutional sector with 2002/03 constant prices
(Seasonally adjusted). The end-year net private capital stock involves both non-financial
and financial corporations. Population data in Australia (000 persons) is obtained mainly
from OECD Economic Outlook: Table C3-AUS-Y (1960–2003). Population level in 1959
originates from World Bank World Tables.
All the per-capita variables are expressed in logarithms as implied by equations (23) and
(24). Not surprisingly, from figure 2, we observe the three series roughly exhibit increasing
time trends in our sample period. However, the net public capital stock per capita (in
logarithm) seems to suffer from a structural change after 1979 and remains roughly constant
thereafter. Therefore, in our empirical estimations and tests, we will take into account this
structural break.
3See Hakkio and Rush (1991). All data were extracted from dX Database.
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 9
FIGURE 1. LOG PER CAPITA OUTPUT, PRIVATE CAPITAL AND PUBLIC CAPITAL, RESPECTIVELY.
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 20052.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
(a) ln y
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 20052.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
(b) ln k
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 20052
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
(c) ln g
10 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
Weak stationarity of the series: From Table 1, the Augmented Dickey-Fuller (1979,
1981) unit root test reveals all the variables appear to be non-stationary in levels (contain
a unit root), but will be stationary after taking first differences.4 Thus, for latter estimation
of cointegrating relationship as well as vector error correction model, we will not result in
imbalanced regressions since the levels of variables are integrated with order 1, or I(1), but
with order 0, or I(0), after taking first differences.
TABLE 1
UNIT ROOT TESTS FOR ln y, ln k AND ln g IN LEVELS AND FIRST DIFFERENCES
Endogenous variables in levelsln y ln k ln g ln g
Sample period 1960–2003 1961–2003 1964–1978 1979–2003Lags of ADF test 0 1 4 2Exogenous constant and trend constant and trend constant and trend constantADF statistics −2.1479 −2.1439 −0.2497 −2.53095% critical value −3.5155 −3.5181 −3.7597 −2.9862
Endogenous variables in first differences∆ ln y ∆ ln k ∆ ln g ∆ ln g
Sample period 1961–2003 1961–2003 1964–1978 1979–2003Lags of ADF test 0 0 3 1Exogenous constant constant constant and trend constantADF statistics −5.6132 −3.0520 −4.5334 −4.70515% critical value −2.9314 −2.9314 −3.7597 −2.9862
Note: the lag length of each dependent variable is optimally determined by Schwarz information criterion (SC).
Cointegration and output elasticity estimates: The long-run cointegrating relation-
ships between ln y, ln k and ln g depends on their non-stationary fluctuation along the time
trend. Firstly, the cointegrating relationships in equation (21) and (22) are rejected as their
linear combination of residuals fail to be a stationary process. Therefore, we cannot have
a maximum of two linearly independent cointegrating vectors in out sample. These results
are shown in Table 2.
Next, we consider whether there exists one cointegrating vector between the three variables
in this system. To motivate this possibility, consider the three-dimensional scatter plot of
{ln y, ln k, ln g} in Figure 2. This raw and informal plot suggests that all three series occur
along some common vector at least in the long run. In other words, there is some informal
evidence of a single cointegrating vector for all three variables. There appears to be a
little kink in the scatter plot (in the “north-eastern” direction) possibly due to the apparent
structural break in the time series for ln g.
As previously mentioned, we allow the structural break of ln g to augment the regression
4We modify the unit root test for ln g by examining two separated period to avoid its structural change after 1979.
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 11
TABLE 2
UNIT ROOT TESTS FOR (ln y − ln k) AND (ln y − ln g)
Endogenous variables in levels(ln y − ln k) (ln y − ln g)
Sample period 1960–2003 1961–2003Lags of ADF test 0 1Exogenous with constant and trend with constantADF statistics −1.4650 0.96745% critical value −3.5155 −2.9134
Note: the lag length of each dependent variable is optimally determined by Schwarz information criterion.
for the theoretical relationship in equations (23) and (24). Therefore, the long-run relationship
between the three series is determined by
ln yt = β0 + β1 ln kt + β2 ln gt + β3t+ β4D79 + β5D79t (25)
where D79 is the dummy variable that captures the obvious structural change in the series
{ln gt} such that
D79 =
{1 if t ∈ {1979, 1980, ..., 2003}0 if t ∈ {1959, 1960, ..., 1978}
and the βi’s are reduced-form coefficients to be estimated.
FIGURE 2. SCATTER PLOT OF LOG PER CAPITA OUTPUT, PRIVATE CAPITAL AND PUBLIC CAPITAL.
2.62.8
33.2
3.43.6
3.84
2
2.5
3
3.5
42
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
ln yln k
ln g
The existence of cointegrating relationship amongst the three variables requires the OLS
12 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
residuals from equation (25) to be stationary otherwise we would have a spurious regression.
The reduced-form coefficients β1, β2 and β3 in equation (25), respectively, denote (α− θφ),
θ and {1 − [α + (1 − φ)θ]x} originating from equation (24). Specifically, if the estimated
coefficients follow the conditions that β1 + β2 = 1 and β3 = 0, the engines of economic
growth are endogenously determined by both net private and public capital (that is, equation
(23) holds such that the production function in equation (3) exhibits constant returns to
scale). Conversely, if β1 + β2 is estimated to be less than one (i.e. β3 > 0 and equation (24)
holds in this system), the economy will grow exogenously along the time trend.
The OLS estimation provides the following coefficients embedded in equation (25) and
the parentheses below estimated coefficients denote respective standard errors.
ln yt = 0.6070(0.4176)
+ 0.0927(0.1885)
ln kt + 0.9520(0.3865)
ln gt − 0.0126(0.0070)
t− 0.6516(0.1868)
D79 + 0.0306(0.0099)
D79t (25’)
From the above estimation results, we cannot reject the null hypotheses of β1 + β2 = 1
and that β3 = 0 for 5% level of significance. This suggests that the hypothesis of strict
endogenous growth due to public capital spillover into private production is the main driving
force in Australia during our sample period. Moreover, the net government capital plays a
more significant role (with very high input share 0.9520) than net private capital does (with
insignificant input share 0.0927). Our result contrasts with the results from Lau and Sin
(1997), who used US data for a similar regression and estimated the shares of ln g and ln k
to be 0.11 and 0.43, respectively. The result of Lau and Sin (1997) suggested the dominance
of private capital in the process of economic growth in the US. Another study for Australia
was carried out by Otto and Voss (1994) using a shorter sample period of 1966/67–1989/90.
Although their findings revealed a higher share of public capital in the production process
than Otto and Voss (1994), the share of private capital was estimated to be unreasonably
negative.
Economic relevance of long-run estimates: Since one of the parameters {α, φ} cannot
be identified from the estimates of β1 and β2 = θ, we perform the following informal exercise
to check whether our estimates provide some sensible economic parameterization. We use
the following guideline. Since we know very little about the congestion parameter φ in the
model, except that it must be constrained within the open set (0, 1), we calibrate α to two
scenarios such that α = 1/3 in one scenario and α = 1/4 in the other. The latter calibration
is motivated by the argument that the usual share of private capital stock in levels to be
lower than the stylized fact of 1/3 in the presence of endogenous growth effects. Table 3
demonstrates a comparison of estimated results between the two previous studies and our
findings. It should noted that when considering the congestion effect of aggregate public
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 13
infrastructure in equation (2), the estimated coefficients of ln g and ln k from Lau and Sin
(1997) seem to result in a theoretically infeasible congestion parameter φ when we set the
private capital share α as a reasonable scale (either 1/3 or 1/4). However, in our calculations
of φ, based on our reduced-form estimates and the assumptions on α, we do not have such
problems.
TABLE 3
THE DIFFERENCES OF ESTIMATED RESULTS BETWEEN THE US AND AUSTRALIA
Lau and Sin (1997) Otto and Voss (1994) This paperEstimated coefficients US data Australia data Australia dataβ1 = (α− θφ) 0.43 −0.0870 0.0927β2 = θ 0.11 0.4303 0.9520
Comparison α = 1/3 α = 1/4 α = 1/3 α = 1/4 α = 1/3 α = 1/4Congestion parameter φ −0.88 −1.63 0.9768 0.7832 0.25 0.17
Using again the Augmented Dickey-Fuller unit root test for the OLS residuals from
regression (25’), we found that they are stationary for 5% level of significance.5 This evidence
suggests that ln y, ln k and ln g are cointegrated with intercept and time trend in equation
(25’).
4. SHORT-RUN DYNAMICS AND IMPULSE RESPONSE
Aschauer (1989a) pointed out the possibility of reverse causation between the level of
public capital expenditure and production. That is, ln g responds to rises in ln y. This example
of Wagner’s Law arises if expenditure on public infrastructure or public goods is a superior
good. Furthermore, there may also be interactions between ln g and ln k. On the one hand,
public capital expenditure may be seen as the springboard for private investment. This runs
counter to standard elementary macroeconomic argument that government expenditure tends
to crowd out private investment. However, it may be that public capital increases the marginal
product of private capital. An obvious example is the provision of better highways, which
results in less wear and tear of private vehicles while goods are transported more efficiently.
On the other hand, public capital expenditure may be seen as responding to private investment
demands.The time-series causal chain running from public capital accumulation is obvious in
the theoretical model. Here we would like to investigate whether this short-run effect is
empirically plausible by letting the data speak as much as possible while imposing the long-
run relationship estimated earlier. Thus we conduct impulse response analysis of a vector
5We also reject the null hypotheses of first- and second-order serial correlation in errors, for 1% level of significance,using Breusch-Godfrey serial correlation LM test. This provides a robustness check for our estimation.
14 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
error correction model with no theoretical restrictions on the short run dynamics to let the
data inform us of the short-run effect of ln g on ln y and ln k.
4.1 Impulse Response Analysis
Our optimally chosen VEC(2) model incorporating (25) with exogenous variables t, D79
and D79t, can be written as a restricted vector autoregression model with lag length 3
(VAR(3)). The restriction arises from the cointegration structure imposed by the long run
behavior. The multivariate error structure from this is decomposed into lower triangular
matrices such that the restricted VAR can be written, theoretically, as an infinite vector
moving average (VMA) utilising the orthogonalised error structure. This then, ensures that
a shock to a variable will have no contemporaneous correlation with other residuals.
In performing an impulse response of VEC(2) model, a simulation period of up to fifty
years and an ordering of (ln y, ln k and ln g) is used. This ordering is determined by
VAR(3) Pairwise Granger Causality/Block Exogeneity Wald Tests. Since the variables are
in logarithms, each 0.01 unit change in the response functions denotes an 1 per cent change.
From Figure 3, ln y responds positively to a one-period (positive) shock from ln g and this
effect fluctuates and diminishes a little over time. The response of ln y to ln g (permanently)
reaches 0.6 per cent level after 40 years. The ln g shock has permanent effects on ln y due
to ln y being close to a unit root.
Also, ln k responds positively to ln g from the beginning and reaches the highest response
above 0.6 per cent. Thus, there is evidence of public infrastructure “crowding in” private
investment, which affirms Aschauer (1989b). Similar to the response of ln y to ln g, the effect
on ln k resulting from a positive shock to ln g is also declining across time and roughly
constant at 0.4% after 40 years. However, there does appear to be evidence of Wagner’s
Law as ln ln g responds positively to ln y.
The influence of ln g can further been seen in the variance decomposition of the fifty-
period forecast error of the variables in the system in Figure 4. It can be observed that
about 30 per cent of the forecast error in ln y is due to the innovation to ln g and about
20 per cent of the forecast error of ln g is due to its own innovation. There contribution
of ln g to the forecast error of ln k, of about 10 per cent, is slightly lower but nevertheless
substantial. Therefore, it can be concluded from the impulse response analysis and variance
decompositions that public infrastructure investment does impact positively on output and
private investment in the short to medium term.
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 15
5. CONCLUSIONS
It was the aim in this paper to study the effect of public infrastructure on the aggregate
economy in terms of long-run growth and short-run effects. In particular, the issue was
whether growth was determined in the long run, in part, by the accumulation of the stock of
public infrastructure. A simple stochastic growth model nesting exogenous and endogenous
growth with public capital spillovers was considered in Section 2 of the paper.
The long-run implication of this model was tested empirically for Australia in Section 3. It
was found that there was evidence of cointegration between per capita output, per capita net
private capital and per capita net public capital. A nested test of the strictly exogenous growth
model was rejected in favour of the endogenous growth model with public infrastructure
spillovers.
Lastly, the cointegrating relationship was incorporated into a VEC(2) model to study the
short-run behaviour of the variables in Section 4. It was found that innovations to public
infrastructure induce permanently higher levels of output and private investment in the short
to medium run.
TIMOTHY KAM, School of Economics & C.A.M.A., Arndt Building 25a, The Australian National University,
ACT 0200, Australia. Email: [email protected].
YI-CHIA WANG, School of Economics, Arndt Building 25a, The Australian National University, ACT 0200,
Australia. E-mail: [email protected].
16 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
FIGURE 3. IMPULSE RESPONSE FUNCTIONS WITH ORDERING (ln y, ln k AND ln g)
-.004
.000
.004
.008
.012
.016
5 10 15 20 25 30 35 40 45 50
Response of lny to lny
-.004
.000
.004
.008
.012
.016
5 10 15 20 25 30 35 40 45 50
Response of lny to lnk
-.004
.000
.004
.008
.012
.016
5 10 15 20 25 30 35 40 45 50
Response of lny to lng
.000
.001
.002
.003
.004
.005
.006
.007
.008
.009
5 10 15 20 25 30 35 40 45 50
Response of lnk to lny
.000
.001
.002
.003
.004
.005
.006
.007
.008
.009
5 10 15 20 25 30 35 40 45 50
Response of lnk to lnk
.000
.001
.002
.003
.004
.005
.006
.007
.008
.009
5 10 15 20 25 30 35 40 45 50
Response of lnk to lng
.000
.001
.002
.003
.004
.005
.006
5 10 15 20 25 30 35 40 45 50
Response of lng to lny
.000
.001
.002
.003
.004
.005
.006
5 10 15 20 25 30 35 40 45 50
Response of lng to lnk
.000
.001
.002
.003
.004
.005
.006
5 10 15 20 25 30 35 40 45 50
Response of lng to lng
Response to Cholesky One S.D. Innovations
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 17
FIGURE 4. FORECAST-ERROR VARIANCE DECOMPOSITIONS WITH ORDERING (ln y, ln k AND ln g)
0
20
40
60
80
100
5 10 15 20 25 30 35 40 45 50
Percent lny variance due to lny
0
20
40
60
80
100
5 10 15 20 25 30 35 40 45 50
Percent lny variance due to lnk
0
20
40
60
80
100
5 10 15 20 25 30 35 40 45 50
Percent lny variance due to lng
0
10
20
30
40
50
60
70
80
5 10 15 20 25 30 35 40 45 50
Percent lnk variance due to lny
0
10
20
30
40
50
60
70
80
5 10 15 20 25 30 35 40 45 50
Percent lnk variance due to lnk
0
10
20
30
40
50
60
70
80
5 10 15 20 25 30 35 40 45 50
Percent lnk variance due to lng
10
20
30
40
50
60
5 10 15 20 25 30 35 40 45 50
Percent lng variance due to lny
10
20
30
40
50
60
5 10 15 20 25 30 35 40 45 50
Percent lng variance due to lnk
10
20
30
40
50
60
5 10 15 20 25 30 35 40 45 50
Percent lng variance due to lng
Variance Decomposition
18 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
APPENDIX A
DYNAMIC PROGRAMMING FOR THE HOUSEHOLD PROBLEM
The method of solving the household’s intertemporal utility maximisation problem subject
to given constraints and public policy in equation (6) is as follows. Bellman’s (1957) principle
of optimality dictates that if the sequence of {ct, kt+1}∞t=0 is maximising, then it must also be
the case that it maximises the functional over {c0, k1} and {ct, kt+1}∞t=1. Hence the problem
in equation (6) can be written as
V (kt, εPt ) = max
ct,kt+1
[ln(ct) + βEtV (kt+1)
]; β ∈ (0, 1) (A.1)
subject to constraints (6) (a)–(c).
A guess of the solution to (A.1) is
V (k0, εP0 ) = B0 +B1 ln(k0) +B2 ln(εP0 ) (A.2)
Substituting the form of equation (A.2) into (A.1) gives
V (kt, εPt ) = max
ct,kt+1
{ln(ct) + βEtV
[B0 +B1 ln(kt+1) +B2 ln(εPt+1)
]}(A.3)
subject to constraints (6) (a)–(c).
At time t, the control variables, ct and kt+1, and the state variables, εPt and kt are all
known. Further, with the assumption that ln εt is independently and identically distributed
such that Et ln(εt+1) = 0, the terms in the curly brackets of equation (A.3) can be reduced
to
ln ct + βEt
[B0 +B1 ln(kt+1)
](A.4)
Define the Lagrangian as
L = ln ct + βEt
[B0 +B1 ln(kt+1)
]+ λt
[(1− τt)Ak
α−θφt gθt ε
Pt − ct − kt+1
]and the first-order conditions for maximisation are
1
ct= λt (A.5)
βB1
kt+1
= λt (A.6)
(1− τt)Akα−θφt gθt ε
Pt = ct + kt+1 (A.7)
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 19
for all states and dates t ∈ N, and the transversality condition
limt→∞
βtkt+1 = 0.
Substitute equations (A.5) and (A.6) into (A.7) to get
ct =1
(1 + βB1)(1− τt)Ak
α−θφt gθt ε
Pt (A.8)
Use the natural constraint (A.7) and (A.8) to derive the stochastic difference equation for
private capital per efficiency unit worker
kt+1 =βB1
(1 + βB1)(1− τt)Ak
α−θφt gθt ε
Pt (A.9)
Substitute equations (A.8) and (A.9) into the RHS of the Bellman equation (A.3) to verify
that the LHS of equation (A.3)
V (kt, εPt ) = B0 +B1 ln(kt) +B2 ln(εPt )
is equal to the RHS of equation (A.3), which is
ln
[1
(1 + βB1)(1− τt)Ak
α−θφt gθt ε
Pt
]+β
{B0 +B1 ln
[βB1
(1 + βB1)(1− τt)Ak
α−θφt gθt ε
Pt
]}.
Expanding terms on the RHS
βB0 − (1 + βB1) ln(1 + βB1) + βB1 ln(βB1) + (1 + βB1) ln[(1− τt)Ag
θt
]+ (α− θφ)(1 + βB1) ln(kt) + (1 + βB1) ln(εPt )
For the functional to be valid, the LHS must, inter alia, satisfy the condition that
B1 = (α− θφ)(1 + βB1)
and thus, the guess in (A.4) will be correct if
B1 =(α− θφ)
1− (α− θφ)β(A.10)
Substitute (A.10) into equations (A.8) and (A.9) then we can obtain the optimal household
consumption and investment paths with given public policy {τt, gt}∞t=0,
ct = (1− τt) [1− (α− θφ)β)]Akα−θφt gθt εPt (7)
kt+1 = (1− τt)(α− θφ)βAkα−θφt gθt εPt (8)
for all states and dates t ∈ N, given k0.
20 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
APPENDIX B
THE GOVERNMENT’S PROBLEM AND OPTIMAL OUTCOMES
The government’s Ramsey optimal fiscal policy problem is defined here as a dynamic
program:
v(kt, gt, εPt ) = max
τt,kt+1,gt+1
{ln{(1− τt) [1− (α− θφ)β)]Akα−θφt gθt ε
Pt }+ βEtv(kt+1, gt+1)
}(B.1)
subject to constraints (10) (a)–(d).
Second, guess that the solution is of the form below
v(k0, g0, εP0 ) = B0 +B1 ln(k0) +B2 ln(g0) +B3 ln(εP0 ) (B.2)
Utilising the guess in (B.2), re-write equation (B.1) as
v(kt, gt, εPt ) = max
τt,kt+1,gt+1
{ln{(1− τt) [1− (α− θφ)β)]Akα−θφt gθt ε
Pt }
+βEt
[B0 +B1 ln(kt+1) +B2 ln(gt+1) +B3 ln(εPt+1)
]}(B.3)
subject to (10) (a)–(d). It is also assumed here that ln(εPt+1) is white noise and therefore,
Et ln(εPt+1) = 0.
Define the Lagrangian as
L ={
ln{(1− τt) [1− (α− θφ)β)]Akα−θφt gθt εPt }
}+ β
[B0 +B1 ln(kt+1) +B2 ln(gt+1)
]+ µt
(τtAk
α−θφt gθt ε
Pt − gt+1
)+ ψt
[(1− τt)(α− θφ)βAkα−θφt gθt ε
Pt − kt+1
](B.4)
The necessary first-order conditions for maximisation of the Lagrangian are
Lkt+1=βB1
kt+1
− ψt = 0 (B.5)
Lgt+1 =βB2
gt+1
− µt = 0 (B.6)
Lτt = − 1
(1− τt)+ µtAk
α−θφt gθt ε
Pt − ψt(α− θφ)βAkα−θφt gθt ε
Pt = 0 (B.7)
Lµt = τtAkα−θφt gθt ε
Pt − gt+1 = 0 (B.8)
Lψt = (1− τt)(α− θφ)βAkα−θφt gθt εPt − kt+1 = 0 (B.9)
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 21
for all states and dates t ∈ N, and the transversality condition
limt→∞
βtkt+1 = 0.
Express the constraints (B.8) and (B.9) in terms of gt+1 and kt+1 respectively, and substitute
these into equations (B.5) and (B.6) to obtain
βB1
(1− τt)(α− θφ)βAkα−θφt gθt εPt
= ψt (B.4’)
βB2
τtAkα−θφt gθt ε
Pt
= µt (B.5’)
Substitute equations (B.4’) and(B.5’) into equation (B.7) gives
τt =βB2
1 + βB1 + βB2
(B.9)
Further substitution of equation (B.9) back into constraints (B.8) and (B.9) results in
gt+1 =
(βB2
1 + βB1 + βB2
)Akα−θφt gθt ε
Pt (B.10)
and
kt+1 = (α− θφ)β
(βB2
1 + βB1 + βB2
)Akα−θφt gθt ε
Pt (B.11)
Next, substitute (B.10) and (B.11) into (B.3) and compare with the form of (B.2)
B0 +B1 ln(kt) +B2 ln(gt) +B3 ln(εPt )
≡ ln
{[1− (α− θφ)β]
(1 + βB1
1 + βB1 + βB2
)Akα−θφt gθt ε
Pt
}+ β
{B0 +B1 ln
[(α− θφ)β
(1 + βB1
1 + βB1 + βB2
)Akα−θφt gθt ε
Pt
]}+ β
{B2 ln
[(α− θφ)β
(βB2
1 + βB1 + βB2
)Akα−θφt gθt ε
Pt
]}(B.12)
Expand the RHS of equation (B.12) and collect terms
βB0 + ln
{[1− (α− θφ)β](1 + βB1)
1 + βB1 + βB2
}+ βB1 ln
[(α− θφ)β(1 + βB1)
1 + βB1 + βB2
]+ βB2 ln
[(α− θφ)β2B2
1 + βB1 + βB2
]+ (1 + βB1 + βB2) ln(A)
+ (α− θφ)(1 + βB1 + βB2) ln(kt)
+ θ(1 + βB1 + βB2) ln(gt) + (1 + βB1 + βB2) ln(εPt ) (B.13)
22 ANU WORKING PAPERS IN ECONOMICS AND ECONOMETRICS NO. 474
For the functional to be valid, that is the LHS=RHS in (B.12), it must be that the coefficients
on the LHS of (B.12) satisfy, inter alia
B1 = (α− θφ)(1 + βB1 + βB2)
B2 = θ(1 + βB1 + βB2)
Solving for B1 and B2 yields
B1 =α− θφ
1− θβ − (α− θφ)β(B.14)
B2 =θ
1− θβ − (α− θφ)β(B.15)
Substitution of equation (B.14) and (B.15) into (B.9), (B.10) and (B.11) gives the optimal
tax rate and the evolutions of private and public capital
τt = θβ (11)
kt+1 = (1− θβ)(α− θφ)βAkα−θφt gθt εPt (12)
gt+1 = θβAkα−θφt gθt εPt (13)
for all states and dates t ∈ N, given k0.
KAM & WANG: PUBLIC CAPITAL SPILLOVERS DOWNUNDER 23
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